2011 v (1) Mathematica, Maple (2) MATLAB (3) CabriGeometry II Plus 1
|
|
|
- とき しげまつ
- 5 years ago
- Views:
Transcription
1 2011 v (1) Mathematica, Maple (2) MATLAB (3) CabriGeometry II Plus 1
2 (1) Maxima Maxima 2006 ISBN (2) Scilab( Home Page) GNU Octave FreeMat (3) KSEG KidsCindy GeoGebra (4) GRAPES (5) TeX 2 Maxima 21 Maxima CD-R maxima maxima-5232exe ( Windows windows version 5232 maxima-5232exeexe 22 Maxima(Windows ) Maxima wxmaxima, xmaxima, maxima (1) wxmaxima maxima-5232 wxmaxima 2
3 (2) (wxmaxima version ) wxmaxima(version 087 ; Shift Enter Enter Enter ; Shift Enter 1/2+1/3 Shift Enter (3) (%i1) (input) (%o1) (output) 1/2+1/3 Shift Enter (%i1) 1/2+1/3; (%o1) 5 6 %o1 % 23 load 3
4 batch diff_exammac f(x) := x^2; diff(f(x), x); g(y) := sin(y); g(f(x)); diff( g(f(x)), x); HTML Maxima y i,j (i = 1, 2, j = 1, 2, 3) P f(i,j)= j i, 3, 3, A (i,j) j i A numer true falsetrue false b-1(= ) 4
5 radcan OK (%i43) example(radcan); (%i44) (log(x+x^2)-log(x))^a/log(1+x)^(a/2) (%o44) (log(x2 + x) log(x)) a log(x + 1) a/2 (%i45) radcan(%) (%o45) log(x + 1) a/2 radcan(%) % (%o44) 24 Maxima (1) + - * / ( ) Mathematica * (x+1)*(x+2), 2*x^2+3*x*y ^ (2) = : y:(x+1)*(x+2) (3) := f(x):=3*x+1 (4) π %pi e %e inf i %i x 4 x^4 x**4 n! n! realpart( ) imagpart( ) sqrt( ) abs( ) exp( ) log( ) sin( ), cos( ), tan( ) asin( ), acos( ), atan( ) sinh( ), cosh( ), tanh( ) (%i1) sqrt(5); (%i4) exp(log(2)); (%o1) 5 (%o4) 2 (%i2) sin(%pi/3); (%i5) (sqrt(7))^2; 3 (%o2) (%o5) 7 2 (%i3) log(%e); (%o3) 1 5
6 (5) ev(n*(n+1)/2 n=10) float(%pi) fpprec:25; bfloat(exp(10)); sum(i^2 i 3 9 ) nusum(i^2, i, 1, n) n=10 n*(n+1)/2 %pi 16 exp(10) 25 b i=3 i 2 (= 280) ni=1 i 2 (= n(n + 1)(2n + 1)/6) limit( (1+n)/n, n, inf ) lim n (1 + n)/n (= 1) (%i1) ev(n*(n+1)/2 n=10); (%i5) sum(i^2 i 3 9 ); (%o1) 55 (%o5) 280 (%i2) float(exp(10)); (%i6) nusum(i^2, i, 1, n); n(n+1)(2 n+1) 6 (%o2) (%o6) (%i3) fpprec:25;bfloat(exp(10)); (%i7) limit((1+n)/n, n, inf); (%o3) 25 (%o7) 1 (%o4) b4 (6) expand( (x+2)^3 ) factor( x^3-1 ) ratsimp( 1/(x-1)+1/(x+1) ) trigsimp(sin(x)^2+cos(x)^2) radcan( ) trigreduce( ) trigexpand(sin(x+y)) (rational function) (trigonometric function) (%i1) (1+sqrt(7))^2; (%i6) 1/(x-1)+1/(x+1); (%o1) ( 7 + 1) 2 1 (%o6) x 1 x+1 (%i2) expand((1+sqrt(7))^2); (%i5) ratsimp(1/(x-1)+1/(x+1)); (%o2) 2 2x (%o5) x 2 1 (%i3) expand((x+2)^2); (%i6) (sin(x))^2+(cos(x))^2; (%o3) x x + 4 (%o6) sin(x) 2 + cos(x) 2 (%i4) factor( ); (%i7) trigsimp((sin(x))^2+(cos(x))^2); (%o4) (%o7) 1 (%i5) factor(x^3-1); (%i8) trigexpand(sin(x+y)); (%o5) (x 1)(x 2 + x + 1) (%o8) cos(x)sin(y)+sin(x)cos(y) (7) 6
7 solve( ) solve([ 1 2,] [ 1 2,]) (%i1) solve(x^2-5*x+6=0,x); (%o1) [x=3,x=2] (%i2) solve([3*x-4*y=10,3*x+2*y=4],[x,y]); (%o2) [[x=2,y=-1]] (8) diff( 2*x^2+3*x*y, x ); x 2 diff( 2*x^2+3*x*y, x, 2 ); x taylor( sin(x), x, 0, 5 ) x x = 0 5 (%i1) f:2*x^2+3*x*y; (%i3) diff(%,y); (%o1) 2x 2 + 3xy (%o3) 3 (%i2) diff(f,x); (%i4) diff(sin(cos(x)),x); (%o2) 3y + 4x (%o4) -sin(x)cos(cos(x)) (%i1) g(x,y):=2*x^2+3*x*y; (%o1) g(x, y) := 2x 2 + 3xy (%i2) diff(g(x,y+x),x); (%o2) 3(y + x) + 7x (9) integrate( 3*x^2+6*x*y, x ); x integrate( 3*x^2+6*x*y, x, 0, 2 ); x 0 2 romberg( 2*cos(x^2+3*x), x, 1, %pi); x 1 π (%i1) integrate(sin(2*x),x); cos(2 x) (%o1) 2 (%i2) integrate(sin(x),x,0,%pi); (%o2) 2 (%i3) romberg(sin(x),x,0,%pi); (%o3) (10) 1 ode2(,, ) %c, %k1, %k2 7
8 1 ode2( diff(y, x)=2*y, y, x) 1 ic1(, xval, yval) 2 ic2(, xval, yval, dval) xval, yval, dval 2 ic1(ode2( diff(y, x)=2*y, y, x), x=0, y=y0) diff y x 3 ic2(ode2( diff(y,x,2)=y,y,x),x=0,y=y0, diff(y,x)=d0) 4 ode2( diff(y, x)=2*y, y, x) ic1(%, x=0, y=y0) (11) desolve(, ) 1 x (t) = 2x(t) desolve( diff(x(t), t, 2)=2*x(t), x(t)) 2 x (t) = y, y (t) = x(t) desolve([ diff(x(t), t)=y(t), diff(y(t), t)=-x(t)],[x(t),y(t)]) atvalue(y(x), xval, yval)) atvalue( diff(y(x),x), xval, dval)) xval, yval, dval 3 y (x) = y(x) y(0) = 1, y (0) = 0 atvalue(y(x), x=0, 1); atvalue( diff(y(x),x), x=0, 0); desolve( diff(y(x),x,2)=-y(x),y(x)); (12) (x1, y1, z1) [x1, y1, z1] [x1, y1, z1][x2, y2, z2] x1 y1 z1 x2 y2 z2 matrix([x1, y1, z1],[x2, y2, z2],[x3, y3, z3]) x3, y3 z3 A A A B 8 transpose(a) A^^-1 AB
9 [x1, y1, z1] 1 x1 [x1, y1, z1][1] sqrt(trigsimp([a*cos(t),a*sin(t)][a*cos(t),a*sin(t)])) [x1, y1, z1], [x1, y1, z1] transpose(adjoint(matrix([x1, y1, z1],[x2, y2, z2],[1,1,1])))[3]; A A : matrix([x1, y1, z1],[x2, y2, z2],[x3, y3, z3]) A (1,2) A[1,2] x = x1 x:matrix([x1],[x2]) x2 Mathematica [ ] ( ) % positive, negative or zero? γ(t) = (a cos(u 0 ) cos(t), a cos(u 0 ) sin(t), a sin(u 0 )) γ (t) = ( a cos(u 0 ) sin(t), a cos(u 0 ) cos(t), 0) s = t 0 γ (t) dt (%i1) gamma:[a*cos(u0)*cos(t),a*cos(u0)*sin(t),a*sin(u0)]; (%o1) [a cos(t)cos(u0),a sin(t)cos(u0),a sin(u0)] (%i2) dgamma:diff(gamma,t); (%o2) [-a sin(t)cos(u0),a cos(t)cos(u0),0] (%i3) s:integrate(sqrt(dgammadgamma),t,0,t); Is a positive or negative? positive Shift Enter Is cos(u0) positive, negative? positive Shift Enter Is t positive or negative or zero? positive Shift Enter (%o3) -a t cos(u0) 25 Maxima Maxima 2 3 gnuplot 3 gnuplot graph option 9
10 2 plot2d(f(x),[x,a,b]) f(x) a x b 3 plot3d(f(x,y),[x,a,b],[y,c,d]) f(x, y) a x b, c y d (1) plot2d(2*x^2,[x,-5,5]); (2) plot2d([05*x^2, x^2, 2*x^2, 3*x^2,4*x^2],[x,-5,5],[y,0,10]); [ ] (3) plot3d(sin(x+sin(y)),[x,-3,3],[y,-3,3]); 21 plot2d([parametric,2*cos(t),sin(t),[t,0,2*%pi]]); plot2d( [parametric, cos(t), sin(t), [t, 0, 2*%pi], [nticks, 50]] ) γ(t) = (cos(t), sin(t)) 0 t 2π plot3d ([cos(x)*(3 + y*cos(x/2)), sin(x)*(3 + y*cos(x/2)), y*sin(x/2)], [x, -%pi, %pi], [y, -1, 1], [grid, 50, 15]); [grid, x, y ] 23 version(5170 ) plot3d([cos(t), sin(t),2*t],[t,-%pi,%pi],[s,-1,1],[grid,50,50]); [s,-1,1] plot3d([cos(t), sin(t),2*t+s],[t,-%pi,%pi],[s,-001,001],[grid,50,50]); 1 gnuplot graph 2 plot2d, plot3d wxplot2d, wxplot3d wxplotsize wxplotsize wxplot2d, wxplot3d wxplotsize:[400,500]; 3 wxmaxima 2 3 (inline) wxplot2d, wxplot3d (1) wxplot3d([u*sin(t),u*cos(t),t/3],[t,0,10],[u,-1,1]); (2) wxplot2d(2*x^2,[x,-5,5]); 10
11 3 KSEG 31 KSEG kseg-0401zip kseg-0401(zip) kseg-0401 C Program Files ) CD-R KSEG kseg-0401 kseg-0401 KSEG(exe) KSeg file Choose Language kseg-0401 kseg_jaqm kseg-0401 kseg_help_ja(html) 32 KSeg (1) 2 (2) (3) (4) (5) (6) A,B,C, C 1,C 2, S 1,S 2, 33 (1) 11
12 (1) Sift (2) BAC BAC Ctrl Z Windows View Pan() Zoom() ( ) Zoom to Fit Original Zoom 100% 12
13 ( ) ( ) ( ) (/) OK () ( ) () () () ( ) () ( ) (1) KSeg ( ) (2) A A Edit/Show label ( / ) 13
14 Edit/Change Label( / ) Edit/Hide Label( / ) (3) A B Edit/Undo( /) Ctrl Z (4) Shift A B 2 (5) New/Segment( / ) AB A,B (6) A Shift B (7) New/Circle By Center And Point( /) A B ( ) (8) B Shift A AB (9) New/Circle By Center And Point( /) New/Circle By Center And Radius( / ) B A (10) Shift (11) New/Intersection Points( / ) 2 View/Zoom To Fit (12) Shift 2 (13) Edit/Hide Object( / ) 2 Edit/Delete Onject( / ) Edit/Unhide All( / ) (14) A C New/Segment ( / ) AC (15) B C New/Segment( / ) BC ABC 1 KSeg 14
15 (16) 3 C,B,A Measure/Angle( / ) CBA 60 CB AB 3 A,B,C (1) 3 A,B,C AB, BC, CA (2) BC ( / ) D (3) CA E (4) AB F (5) 2 A,D AD (6) 2 B,E BE (7) 2 C,F CF (8) 3 AD, BE, CF 1 ( ) 33 (1) 3 A,B,C AB, BC, CA (2) A BC / A BC (3) B CA (4) C AB (5) (1) 3 A,B,C AB, BC, CA (2) BC ( / ) D CA E AB F (3) BC D ( / ) D BC (4) CA E E CA (5) AB F F AB (6) 3 1 ( ) (7) 2 ( / ) (8) (7) A B,C G 35 (1) 3 A,B,C AB, BC, CA 15
16 (2) 3 A,B,C / 2 2 ABC 2 (3) 3 B,C,A BCA 2 (4) 3 C,A,B CAB 2 (5) ABC 3 36 Edit/Hide Object( / ) 37 (0) ABC (1) / ) (2) D,E / (3) ABC / A A ABC (4) A A (5) 38 (0) ABC (1) / ) (2) 3 D,E,F Shift 3 / (3) / (4) ABC / (5) 16
17 (6) (1) (2) Shift 3 / (180 ) (3) / (4) / (0) ABC (1) / ) (2) / (4) ABC / (5) (6) : 1 (1) (5) (6) (2) (7) x y (3) (8) (4) (9) OK 312 (5) (9) (5) (6) (7)/ (8) / (9) OK 17
18 36 Ctrl Ctrl Ctrl Ctrl 313 (1) 2 A,B ) A,B () / A B AB (2) AB C AC (3) DE F (4) F AC (5) D DE D AC DE F G (6) G DE (3) 1 H (7) 2 C,H / (8) F AC AC 314 (1) A B C (2) A (3) D (4) CD E 18
19 (5) CD E CD (6) (5) D / 315 (6) (6) D (7) (5) (6) F (8) D F CF=FD (1) AB C (2) 2 A,C AC CB (3) 2 D,E DE<AB (4) AC D CB E (5) 2 1 P (6) 2 C,P (7) D 317 (1) 2 A B (2) A B (3) C D (4) CD E (5) CD E CD (6) (5) D (7) C Deltoid (1) AB O OA (2) C (3) C,O,A / COA (4) O / (5) B D ( DOB = 2 COA) (6) O,C / 19
20 (7) C / A A C OC E View/Zoom (8) E EC (9) CD (8) C F (10) CD F C 320 Astroid 4a O 1 A a O 2 O 2 O 1 2 Q O 2 P A=Q P=A Q O 2 4 QO 1 A P 321 (1) 2 A,B AB (2) A,B 2 x y (3) x P P AB (4) (3) y Q (5) 2 P,Q 8 (6) 322 (1) (5) (1) A,B A AB (2) (1) P Q PQ (3) P B C P (1) C A P B C Q D 20
21 (4) Q B D / PC QD (5) C PC D QD R P (1) R 37 (construction) (1) File/New Construction (/) (2) 3 A,B,C (3) A,B,C Construction/Make Given ( / ) (Given) (4) AB BC D,E (5) D AB E BC F (6) F A (7) A,B,C Edit/Hide Objects ( / ) A,B,C (8) File/Save as (/) (9) File/New Sketch (/) 3 3 Play/Untitled ( / ) ( sec) (9) 3 (3) P 1, P 2, P 3,, Q 1, Q 2, Q 3, P 1, Q 1 P 2, Q 2 P 3, Q 3 (1) (2) 4 A,B,C,D (3) 2 A,C AC 21
22 (4) C AC ( E ) (5) (6) 2 B,D BD (7) D BD ( F ) (8) (9) 2 E,F (10) Shift 4 A,B,C,D (11) (12) Shift 4 C,D,E,F (13) (14) (15) 2 (l, m (16) l 2 A,C (17) m 2 B,D (18) Shift 4 A,B,C,D / 325 (1) File/New Construction (/) (2) 3 A,B,C (3) AB BC D,E (4) DE F (5) A,B,C () Construction/Make Given ( /) (Given) (6) A,D,F () Construction/Recurse ( / ) (7) F,E,C () Construction/Recurse ( / ) (8) 4 A,B,C,F Edit/Hide Objects ( / ) (9) File/Save as (/) ( sec) (10) ( File/New Sketch (/) ) 3 3 Play ( ) (6)(8) (5) 22
23 (5)(6)(7) 4 Id (1) Ctrl Alt Delete (2) Id Enter (3) Maxima(wxMaxima) maxima-5232 wxmaxima KSeg kseg-0401 KSEG(exe) KSeg file Choose Language C apps kseg-0401 kseg_jaqm KSeg KSeg (1) (2) 5 KNOPPIX/Math KNOPPIX/Math Klaus Knopper KNOPPIX (Kile, platex2e, AMS-TeX, AMS-LaTeX, Prosper (Axiom, Maxima, Risa/Asir(OpenXM) (Geomview, Qhull, Surf, Surface Evolver, XaoS, Yorick (Kseg, Geogebra) (GAP (PARI/GP (Singular, Macaulay2 (SnapPea (R, XLISP-STAT 23
24 (C, C++, Java, Fortran, Ruby, Perl, Python, Scheme, (ATLAS, BLAS, LAPACK, Octave Microsoft Office Open Office KNOPPIX/Math DVD OS Linux KNOPPIX/Math DVD KNOPPIX/Math Windows 24
(R, XLISP-STAT (C, C++, Java, Fortran, Ruby, Perl, Python, Scheme,... Microsoft Office Open Office 1.2 KNOPPIX/Math DVD (1) DVD KNOPPIX/Math DVD Windo
2010 8 4 1 KNOPPIX/Math KNOPPIX/Math Klaus Knopper KNOPPIX(OS Linux) KNOPPIX/Math http://www.knoppix-math.org/wiki/ Windows KNOPPIX/Math http://www.math.kobe-u.ac.jp/vmkm/ KNOPPIX/Math DVD KNOPPIX/Math
. KNOPPIX cloop, 700MB CD 1.8GB. DVD, KNOPPIX DVD 12GB.,.,,,. *2, KNOPPIX. 2.1 Windows Mac CD/DVD, Windows PC KNOPPIX. Apple Mac,, Mac Intel CPU
KNOPPIX/Math KNOPPIX/Math Project 1 Maxima [1],,,.., KNOPPIX/Math *1. CD/DVD PC,. KNOPPIX/Math/2007 2000, 100. KNOPPIX/Math,,. KNOPPIX/Math.,,., CD/DVD,. USB,. KNOPPIX/Math,. 2 KNOPPIX KNOPPIX/Math KNOPPIX
種々の数学ソフトウェア
January 17, 2013 Maple, Maxima, GeoGebra,,, Knoppix/Math Maple,, : Student[Calculus1] : 30 ( 2 ) Maple Doc Maple ( ) cfep, ; ab a*b, x n xˆn x n + x m, xˆn, x n+xm Maple ( ) =, =!= f(x):=xˆ3-3*x+1;
いつでも どこでも スマホで数学! サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 初版 1 刷発行当時のものです.
いつでも どこでも スマホで数学! サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/001201 このサンプルページの内容は, 初版 1 刷発行当時のものです. 0 i Maxima on Android SNS Web Maxima Android Linux, Windows, MacOS
( )
18 10 01 ( ) 1 2018 4 1.1 2018............................... 4 1.2 2018......................... 5 2 2017 7 2.1 2017............................... 7 2.2 2017......................... 8 3 2016 9 3.1 2016...............................
熊本県数学問題正解
00 y O x Typed by L A TEX ε ( ) (00 ) 5 4 4 ( ) http://www.ocn.ne.jp/ oboetene/plan/. ( ) (009 ) ( ).. http://www.ocn.ne.jp/ oboetene/plan/eng.html 8 i i..................................... ( )0... (
D xy D (x, y) z = f(x, y) f D (2 ) (x, y, z) f R z = 1 x 2 y 2 {(x, y); x 2 +y 2 1} x 2 +y 2 +z 2 = 1 1 z (x, y) R 2 z = x 2 y
5 5. 2 D xy D (x, y z = f(x, y f D (2 (x, y, z f R 2 5.. z = x 2 y 2 {(x, y; x 2 +y 2 } x 2 +y 2 +z 2 = z 5.2. (x, y R 2 z = x 2 y + 3 (2,,, (, 3,, 3 (,, 5.3 (. (3 ( (a, b, c A : (x, y, z P : (x, y, x
topics KNOPPIX/Math DVD maxima KSEG GeoGebra Coq 2011 July 17 - Page 1
KNOPPIX/Math ( ) 2011/07/17 topics KNOPPIX/Math DVD maxima KSEG GeoGebra Coq 2011 July 17 - Page 1 (1) 2011 July 17 - Page 2 (2) 2011 July 17 - Page 3 KNOPPIX/Math KNOPPIX: Linux CD/DVD CD/DVD 1 Windows
. UNIX, Linux, KNOPPIX. C,.,., ( 1 ) p. 2
2009 ( 1 ) 2009 ( 1 ) p. 1 . UNIX, Linux, KNOPPIX. C,.,.,. 2009 ( 1 ) p. 2 , +, ( ), ( ), or PC orange2, knxm2008vm, iyokan-6 KNOPPIX/Math (DVD ) 2009 ( 1 ) p. 3 ,. Mathematica (20-30 /1 ), Maple (20 /1
, x R, f (x),, df dx : R R,, f : R R, f(x) ( ).,, f (a) d f dx (a), f (a) d3 f dx 3 (a),, f (n) (a) dn f dx n (a), f d f dx, f d3 f dx 3,, f (n) dn f
,,,,.,,,. R f : R R R a R, f(a + ) f(a) lim 0 (), df dx (a) f (a), f(x) x a, f (a), f(x) x a ( ). y f(a + ) y f(x) f(a+) f(a) f(a + ) f(a) f(a) x a 0 a a + x 0 a a + x y y f(x) 0 : 0, f(a+) f(a)., f(x)
OABC OA OC 4, OB, AOB BOC COA 60 OA a OB b OC c () AB AC () ABC D OD ABC OD OA + p AB + q AC p q () OABC 4 f(x) + x ( ), () y f(x) P l 4 () y f(x) l P
4 ( ) ( ) ( ) ( ) 4 5 5 II III A B (0 ) 4, 6, 7 II III A B (0 ) ( ),, 6, 8, 9 II III A B (0 ) ( [ ] ) 5, 0, II A B (90 ) log x x () (a) y x + x (b) y sin (x + ) () (a) (b) (c) (d) 0 e π 0 x x x + dx e
40 6 y mx x, y 0, 0 x 0. x,y 0,0 y x + y x 0 mx x + mx m + m m 7 sin y x, x x sin y x x. x sin y x,y 0,0 x 0. 8 x r cos θ y r sin θ x, y 0, 0, r 0. x,
9.. x + y + 0. x,y, x,y, x r cos θ y r sin θ xy x y x,y 0,0 4. x, y 0, 0, r 0. xy x + y r 0 r cos θ sin θ r cos θ sin θ θ 4 y mx x, y 0, 0 x 0. x,y 0,0 x x + y x 0 x x + mx + m m x r cos θ 5 x, y 0, 0,
sin x
Mathematica 1998 7, 2001 3 Mathematica Mathematica 1 Mathematica 2 2 Mathematica 3 3 4 4 7 5 8 6 10 7 13 8 17 9 18 10 20 11 21 12 23 1 13 23 13.1............................ 24 13.2..........................
8 i, III,,,, III,, :!,,,, :!,,,,, 4:!,,,,,,!,,,, OK! 5:!,,,,,,,,,, OK 6:!, 0, 3:!,,,,! 7:!,,,,,, ii,,,,,, ( ),, :, ( ), ( ), :... : 3 ( )...,, () : ( )..., :,,, ( ), (,,, ),, (ϵ δ ), ( ), (ˆ ˆ;),,,,,,!,,,,.,,
1 1 Gnuplot gnuplot Windows gnuplot gp443win32.zip gnuplot binary, contrib, demo, docs, license 5 BUGS, Chang
Gnuplot で微分積分 2011 年度前期 数学解析 I 講義資料 (2011.6.24) 矢崎成俊 ( 宮崎大学 ) 1 1 Gnuplot gnuplot http://www.gnuplot.info/ Windows gnuplot 2011 6 22 4.4.3 gp443win32.zip gnuplot binary, contrib, demo, docs, license 5
II 1 3 2 5 3 7 4 8 5 11 6 13 7 16 8 18 2 1 1. x 2 + xy x y (1 lim (x,y (1,1 x 1 x 3 + y 3 (2 lim (x,y (, x 2 + y 2 x 2 (3 lim (x,y (, x 2 + y 2 xy (4 lim (x,y (, x 2 + y 2 x y (5 lim (x,y (, x + y x 3y
t θ, τ, α, β S(, 0 P sin(θ P θ S x cos(θ SP = θ P (cos(θ, sin(θ sin(θ P t tan(θ θ 0 cos(θ tan(θ = sin(θ cos(θ ( 0t tan(θ
4 5 ( 5 3 9 4 0 5 ( 4 6 7 7 ( 0 8 3 9 ( 8 t θ, τ, α, β S(, 0 P sin(θ P θ S x cos(θ SP = θ P (cos(θ, sin(θ sin(θ P t tan(θ θ 0 cos(θ tan(θ = sin(θ cos(θ ( 0t tan(θ S θ > 0 θ < 0 ( P S(, 0 θ > 0 ( 60 θ
2 1 Mathematica Mathematica Mathematica Mathematica Windows Mac *1 1.1 1.1 Mathematica 9-1 Expand[(x + y)^7] (x + y) 7 x y Shift *1 Mathematica 1.12
![2 1 Mathematica Mathematica Mathematica Mathematica Windows Mac *1 1.1 1.1 Mathematica 9-1 Expand[(x + y)^7] (x + y) 7 x y Shift *1 Mathematica 1.12](/thumbs/39/20083158.jpg "2 1 Mathematica Mathematica Mathematica Mathematica Windows Mac *1 1.1 1.1 Mathematica 9-1 Expand[(x + y)^7] (x + y) 7 x y Shift *1 Mathematica 1.12")
Chapter 1 Mathematica Mathematica Mathematica 1.1 Mathematica Mathematica (Wolfram Research) Windows, Mac OS X, Linux OS Mathematica 88 2012 11 9 2 Mathematica 2 1.2 Mathematica Mathematica 2 1 Mathematica
( ) a, b c a 2 + b 2 = c 2. 2 1 2 2 : 2 2 = p q, p, q 2q 2 = p 2. p 2 p 2 2 2 q 2 p, q (QED)
rational number p, p, (q ) q ratio 3.14 = 3 + 1 10 + 4 100 ( ) a, b c a 2 + b 2 = c 2. 2 1 2 2 : 2 2 = p q, p, q 2q 2 = p 2. p 2 p 2 2 2 q 2 p, q (QED) ( a) ( b) a > b > 0 a < nb n A A B B A A, B B A =
function2.pdf
2... 1 2009, http://c-faculty.chuo-u.ac.jp/ nishioka/ 2 11 38 : 5) i) [], : 84 85 86 87 88 89 1000 ) 13 22 33 56 92 147 140 120 100 80 60 40 20 1 2 3 4 5 7.1 7 7.1 1. *1 e = 2.7182 ) fx) e x, x R : 7.1)
18 ( ) I II III A B C(100 ) 1, 2, 3, 5 I II A B (100 ) 1, 2, 3 I II A B (80 ) 6 8 I II III A B C(80 ) 1 n (1 + x) n (1) n C 1 + n C
8 ( ) 8 5 4 I II III A B C( ),,, 5 I II A B ( ),, I II A B (8 ) 6 8 I II III A B C(8 ) n ( + x) n () n C + n C + + n C n = 7 n () 7 9 C : y = x x A(, 6) () A C () C P AP Q () () () 4 A(,, ) B(,, ) C(,,
70 : 20 : A B (20 ) (30 ) 50 1
70 : 0 : A B (0 ) (30 ) 50 1 1 4 1.1................................................ 5 1. A............................................... 6 1.3 B............................................... 7 8.1 A...............................................
(, Goo Ishikawa, Go-o Ishikawa) ( ) 1
(, Goo Ishikawa, Go-o Ishikawa) ( ) 1 ( ) ( ) ( ) G7( ) ( ) ( ) () ( ) BD = 1 DC CE EA AF FB 0 0 BD DC CE EA AF FB =1 ( ) 2 (geometry) ( ) ( ) 3 (?) (Topology) ( ) DNA ( ) 4 ( ) ( ) 5 ( ) H. 1 : 1+ 5 2
no35.dvi
p.16 1 sin x, cos x, tan x a x a, a>0, a 1 log a x a III 2 II 2 III III [3, p.36] [6] 2 [3, p.16] sin x sin x lim =1 ( ) [3, p.42] x 0 x ( ) sin x e [3, p.42] III [3, p.42] 3 3.1 5 8 *1 [5, pp.48 49] sin
(1) (2) (3) (4) HB B ( ) (5) (6) (7) 40 (8) (9) (10)
2017 12 9 4 1 30 4 10 3 1 30 3 30 2 1 30 2 50 1 1 30 2 10 (1) (2) (3) (4) HB B ( ) (5) (6) (7) 40 (8) (9) (10) (1) i 23 c 23 0 1 2 3 4 5 6 7 8 9 a b d e f g h i (2) 23 23 (3) 23 ( 23 ) 23 x 1 x 2 23 x
4 4 4 a b c d a b A c d A a da ad bce O E O n A n O ad bc a d n A n O 5 {a n } S n a k n a n + k S n a a n+ S n n S n n log x x {xy } x, y x + y 7 fx
4 4 5 4 I II III A B C, 5 7 I II A B,, 8, 9 I II A B O A,, Bb, b, Cc, c, c b c b b c c c OA BC P BC OP BC P AP BC n f n x xn e x! e n! n f n x f n x f n x f k x k 4 e > f n x dx k k! fx sin x cos x tan
/02/18
3 09/0/8 i III,,,, III,?,,,,,,,,,,,,,,,,,,,,?,?,,,,,,,,,,,,,,!!!,? 3,,,, ii,,,!,,,, OK! :!,,,, :!,,,,,, 3:!,, 4:!,,,, 5:!,,! 7:!,,,,, 8:!,! 9:!,,,,,,,,, ( ),, :, ( ), ( ), 6:!,,, :... : 3 ( )... iii,,
29
9 .,,, 3 () C k k C k C + C + C + + C 8 + C 9 + C k C + C + C + C 3 + C 4 + C 5 + + 45 + + + 5 + + 9 + 4 + 4 + 5 4 C k k k ( + ) 4 C k k ( k) 3 n( ) n n n ( ) n ( ) n 3 ( ) 3 3 3 n 4 ( ) 4 4 4 ( ) n n
18 ( ) ( ) [ ] [ ) II III A B (120 ) 1, 2, 3, 5, 6 II III A B (120 ) ( ) 1, 2, 3, 7, 8 II III A B (120 ) ( [ ]) 1, 2, 3, 5, 7 II III A B (
![18 ( ) ( ) [ ] [ ) II III A B (120 ) 1, 2, 3, 5, 6 II III A B (120 ) ( ) 1, 2, 3, 7, 8 II III A B (120 ) ( [ ]) 1, 2, 3, 5, 7 II III A B (](/thumbs/100/143999768.jpg "18 ( ) ( ) [ ] [ ) II III A B (120 ) 1, 2, 3, 5, 6 II III A B (120 ) ( ) 1, 2, 3, 7, 8 II III A B (120 ) ( [ ]) 1, 2, 3, 5, 7 II III A B (")
8 ) ) [ ] [ ) 8 5 5 II III A B ),,, 5, 6 II III A B ) ),,, 7, 8 II III A B ) [ ]),,, 5, 7 II III A B ) [ ] ) ) 7, 8, 9 II A B 9 ) ) 5, 7, 9 II B 9 ) A, ) B 6, ) l ) P, ) l A C ) ) C l l ) π < θ < π sin
i
i 3 4 4 7 5 6 3 ( ).. () 3 () (3) (4) /. 3. 4/3 7. /e 8. a > a, a = /, > a >. () a >, a =, > a > () a > b, a = b, a < b. c c n a n + b n + c n 3c n..... () /3 () + (3) / (4) /4 (5) m > n, a b >, m > n,
入試の軌跡
4 y O x 4 Typed by L A TEX ε ) ) ) 6 4 ) 4 75 ) http://kumamoto.s.xrea.com/plan/.. PDF) Ctrl +L) Ctrl +) Ctrl + Ctrl + ) ) Alt + ) Alt + ) ESC. http://kumamoto.s.xrea.com/nyusi/kumadai kiseki ri i.pdf
1 26 ( ) ( ) 1 4 I II III A B C (120 ) ( ) 1, 5 7 I II III A B C (120 ) 1 (1) 0 x π 0 y π 3 sin x sin y = 3, 3 cos x + cos y = 1 (2) a b c a +
6 ( ) 6 5 ( ) 4 I II III A B C ( ) ( ), 5 7 I II III A B C ( ) () x π y π sin x sin y =, cos x + cos y = () b c + b + c = + b + = b c c () 4 5 6 n ( ) ( ) ( ) n ( ) n m n + m = 555 n OAB P k m n k PO +
Fortran90/95 [9]! (1 ) " " 5 "Hello!"! 3. (line) Fortran Fortran 1 2 * (1 ) 132 ( ) * 2 ( Fortran ) Fortran ,6 (continuation line) 1
![Fortran90/95 [9]! (1 ) 5 Hello!! 3. (line) Fortran Fortran 1 2 * (1 ) 132 ( ) * 2 ( Fortran ) Fortran ,6 (continuation line) 1](/thumbs/48/24845697.jpg "Fortran90/95 [9]! (1 ) 5 Hello!! 3. (line) Fortran Fortran 1 2 * (1 ) 132 ( ) * 2 ( Fortran ) Fortran ,6 (continuation line) 1")
Fortran90/95 2.1 Fortran 2-1 Hello! 1 program example2_01! end program 2! first test program ( ) 3 implicit none! 4 5 write(*,*) "Hello!"! write Hello! 6 7 stop! 8 end program example2_01 1 program 1!
0.6 A = ( 0 ),. () A. () x n+ = x n+ + x n (n ) {x n }, x, x., (x, x ) = (0, ) e, (x, x ) = (, 0) e, {x n }, T, e, e T A. (3) A n {x n }, (x, x ) = (,
[ ], IC 0. A, B, C (, 0, 0), (0,, 0), (,, ) () CA CB ACBD D () ACB θ cos θ (3) ABC (4) ABC ( 9) ( s090304) 0. 3, O(0, 0, 0), A(,, 3), B( 3,, ),. () AOB () AOB ( 8) ( s8066) 0.3 O xyz, P x Q, OP = P Q =
GraphicsWithPlotFull.nb Plot[{( 1), ( ),...}, {( ), ( ), ( )}] Plot Plot Cos x Sin x, x, 5 Π, 5 Π, AxesLabel x, y x 1 Plot AxesLabel
![GraphicsWithPlotFull.nb Plot[{( 1), ( ),...}, {( ), ( ), ( )}] Plot Plot Cos x Sin x, x, 5 Π, 5 Π, AxesLabel x, y x 1 Plot AxesLabel](/thumbs/81/83602176.jpg "GraphicsWithPlotFull.nb Plot[{( 1), ( ),...}, {( ), ( ), ( )}] Plot Plot Cos x Sin x, x, 5 Π, 5 Π, AxesLabel x, y x 1 Plot AxesLabel")
http://yktlab.cis.k.hosei.ac.jp/wiki/ 1(Plot) f x x x 1 1 x x ( )[( 1)_, ( )_, ( 3)_,...]=( ) Plot Plot f x, x, 5, 3 15 10 5 Plot[( ), {( ), ( ), ( )}] D g x x 3 x 3 Plot f x, g x, x, 10, 8 00 100 10 5
gnuplot gnuplot 1 3 y = x 3 + 3x 2 2 y = sin x sin(x) x*x*x+3*x*x
gnuplot gnuplot y = x + x y = sin x.8 sin(x) 8 7 6 x*x*x+*x*x.6.. -. -. -.6 -.8 - - - - - - - -. - -. - -.. gnuplot gnuplot> set xrange[-.:.] gnuplot> plot x**+*x** y = x x gnuolot> reset gnuplot> plot
< 1 > (1) f 0 (a) =6a ; g 0 (a) =6a 2 (2) y = f(x) x = 1 f( 1) = 3 ( 1) 2 =3 ; f 0 ( 1) = 6 ( 1) = 6 ; ( 1; 3) 6 x =1 f(1) = 3 ; f 0 (1) = 6 ; (1; 3)
< 1 > (1) f 0 (a) =6a ; g 0 (a) =6a 2 (2) y = f(x) x = 1 f( 1) = 3 ( 1) 2 =3 ; f 0 ( 1) = 6 ( 1) = 6 ; ( 1; 3) 6 x =1 f(1) = 3 ; f 0 (1) = 6 ; (1; 3) 6 y = g(x) x = 1 g( 1) = 2 ( 1) 3 = 2 ; g 0 ( 1) =
1 B64653 1 1 3.1....................................... 3.......................... 3..1.............................. 4................................ 4..3.............................. 5..4..............................
211 kotaro@math.titech.ac.jp 1 R *1 n n R n *2 R n = {(x 1,..., x n ) x 1,..., x n R}. R R 2 R 3 R n R n R n D D R n *3 ) (x 1,..., x n ) f(x 1,..., x n ) f D *4 n 2 n = 1 ( ) 1 f D R n f : D R 1.1. (x,
A(6, 13) B(1, 1) 65 y C 2 A(2, 1) B( 3, 2) C 66 x + 2y 1 = 0 2 A(1, 1) B(3, 0) P 67 3 A(3, 3) B(1, 2) C(4, 0) (1) ABC G (2) 3 A B C P 6
1 1 1.1 64 A6, 1) B1, 1) 65 C A, 1) B, ) C 66 + 1 = 0 A1, 1) B, 0) P 67 A, ) B1, ) C4, 0) 1) ABC G ) A B C P 64 A 1, 1) B, ) AB AB = 1) + 1) A 1, 1) 1 B, ) 1 65 66 65 C0, k) 66 1 p, p) 1 1 A B AB A 67
DVIOUT-MTT元原
TI-92 -MTT-Mathematics Thinking with Technology MTT ACTIVITY Discussion 1 1 1.1 v t h h = vt 1 2 gt2 (1.1) xy (5, 0) 20m/s [1] Mode Graph Parametric [2] Y= [3] Window [4] Graph 1.1: Discussion 2 Window
I A A441 : April 21, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka ) Google
I4 - : April, 4 Version :. Kwhir, Tomoki TA (Kondo, Hirotk) Google http://www.mth.ngoy-u.c.jp/~kwhir/courses/4s-biseki.html pdf 4 4 4 4 8 e 5 5 9 etc. 5 6 6 6 9 n etc. 6 6 6 3 6 3 7 7 etc 7 4 7 7 8 5 59
ORIGINAL TEXT I II A B 1 4 13 21 27 44 54 64 84 98 113 126 138 146 165 175 181 188 198 213 225 234 244 261 268 273 2 281 I II A B 292 3 I II A B c 1 1 (1) x 2 + 4xy + 4y 2 x 2y 2 (2) 8x 2 + 16xy + 6y 2
1990 IMO 1990/1/15 1:00-4:00 1 N N N 1, N 1 N 2, N 2 N 3 N 3 2 x x + 52 = 3 x x , A, B, C 3,, A B, C 2,,,, 7, A, B, C
0 9 (1990 1999 ) 10 (2000 ) 1900 1994 1995 1999 2 SAT ACT 1 1990 IMO 1990/1/15 1:00-4:00 1 N 1990 9 N N 1, N 1 N 2, N 2 N 3 N 3 2 x 2 + 25x + 52 = 3 x 2 + 25x + 80 3 2, 3 0 4 A, B, C 3,, A B, C 2,,,, 7,
名古屋工業大の数学 2000 年 ~2015 年 大学入試数学動画解説サイト
名古屋工業大の数学 年 ~5 年 大学入試数学動画解説サイト http://mathroom.jugem.jp/ 68 i 4 3 III III 3 5 3 ii 5 6 45 99 5 4 3. () r \= S n = r + r + 3r 3 + + nr n () x > f n (x) = e x + e x + 3e 3x + + ne nx f(x) = lim f n(x) lim
210 資料 TI 89 (1) TI 89 2nd ON HOME ( ) ( ) HOME =! ENTER ( ) = (10) ENTER ( ) [ ] { } ( )! 2 =! ( ) ( ) 2 3x ( 2y + yz) ( ) 3x ( ( ) 2y + y z)
![210 資料 TI 89 (1) TI 89 2nd ON HOME ( ) ( ) HOME =! ENTER ( ) = (10) ENTER ( ) [ ] { } ( )! 2 =! ( ) ( ) 2 3x ( 2y + yz) ( ) 3x ( ( ) 2y + y z)](/thumbs/95/123380956.jpg "210 資料 TI 89 (1) TI 89 2nd ON HOME ( ) ( ) HOME =! ENTER ( ) = (10) ENTER ( ) [ ] { } ( )! 2 =! ( ) ( ) 2 3x ( 2y + yz) ( ) 3x ( ( ) 2y + y z)")
210 資料 TI 89 (1) TI 89 2nd ON HOME () () HOME =! ENTER () = 3 10 3 (10) ENTER ( ) [ ] { } ( )! 2 =! ( ) () 2 3x ( 2y + yz) ( ) 3x ( ( ) 2y + y z) ENTER () 2nd 9 2nd 9) ENTER ( ) 2nd 7) ENTER 7 7 ) ENTER
2009 IA 5 I 22, 23, 24, 25, 26, (1) Arcsin 1 ( 2 (4) Arccos 1 ) 2 3 (2) Arcsin( 1) (3) Arccos 2 (5) Arctan 1 (6) Arctan ( 3 ) 3 2. n (1) ta
009 IA 5 I, 3, 4, 5, 6, 7 6 3. () Arcsin ( (4) Arccos ) 3 () Arcsin( ) (3) Arccos (5) Arctan (6) Arctan ( 3 ) 3. n () tan x (nπ π/, nπ + π/) f n (x) f n (x) fn (x) Arctan x () sin x [nπ π/, nπ +π/] g n
I, II 1, A = A 4 : 6 = max{ A, } A A 10 10%
1 2006.4.17. A 3-312 tel: 092-726-4774, e-mail: hara@math.kyushu-u.ac.jp, http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html Office hours: B A I ɛ-δ ɛ-δ 1. 2. A 1. 1. 2. 3. 4. 5. 2. ɛ-δ 1. ɛ-n
(4) P θ P 3 P O O = θ OP = a n P n OP n = a n {a n } a = θ, a n = a n (n ) {a n } θ a n = ( ) n θ P n O = a a + a 3 + ( ) n a n a a + a 3 + ( ) n a n
3 () 3,,C = a, C = a, C = b, C = θ(0 < θ < π) cos θ = a + (a) b (a) = 5a b 4a b = 5a 4a cos θ b = a 5 4 cos θ a ( b > 0) C C l = a + a + a 5 4 cos θ = a(3 + 5 4 cos θ) C a l = 3 + 5 4 cos θ < cos θ < 4
7. y fx, z gy z gfx dz dx dz dy dy dx. g f a g bf a b fa 7., chain ule Ω, D R n, R m a Ω, f : Ω R m, g : D R l, fω D, b fa, f a g b g f a g f a g bf a
9 203 6 7 WWW http://www.math.meiji.ac.jp/~mk/lectue/tahensuu-203/ 2 8 8 7. 7 7. y fx, z gy z gfx dz dx dz dy dy dx. g f a g bf a b fa 7., chain ule Ω, D R n, R m a Ω, f : Ω R m, g : D R l, fω D, b fa,
No2 4 y =sinx (5) y = p sin(2x +3) (6) y = 1 tan(3x 2) (7) y =cos 2 (4x +5) (8) y = cos x 1+sinx 5 (1) y =sinx cos x 6 f(x) = sin(sin x) f 0 (π) (2) y
No1 1 (1) 2 f(x) =1+x + x 2 + + x n, g(x) = 1 (n +1)xn + nx n+1 (1 x) 2 x 6= 1 f 0 (x) =g(x) y = f(x)g(x) y 0 = f 0 (x)g(x)+f(x)g 0 (x) 3 (1) y = x2 x +1 x (2) y = 1 g(x) y0 = g0 (x) {g(x)} 2 (2) y = µ
II A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka )
II 214-1 : October 2, 214 Version : 1.1 Kawahira, Tomoki TA (Kondo, Hirotaka ) http://www.math.nagoya-u.ac.jp/~kawahira/courses/14w-biseki.html pdf 1 2 1 9 1 16 1 23 1 3 11 6 11 13 11 2 11 27 12 4 12 11
高校生の就職への数学II
II O Tped b L A TEX ε . II. 3. 4. 5. http://www.ocn.ne.jp/ oboetene/plan/ 7 9 i .......................................................................................... 3..3...............................
2014 S hara/lectures/lectures-j.html r 1 S phone: ,
14 S1-1+13 http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html r 1 S1-1+13 14.4.11. 19 phone: 9-8-4441, e-mail: hara@math.kyushu-u.ac.jp Office hours: 1 4/11 web download. I. 1. ϵ-δ 1. 3.1, 3..
.5 z = a + b + c n.6 = a sin t y = b cos t dy d a e e b e + e c e e e + e 3 s36 3 a + y = a, b > b 3 s363.7 y = + 3 y = + 3 s364.8 cos a 3 s365.9 y =,
[ ] IC. r, θ r, θ π, y y = 3 3 = r cos θ r sin θ D D = {, y ; y }, y D r, θ ep y yddy D D 9 s96. d y dt + 3dy + y = cos t dt t = y = e π + e π +. t = π y =.9 s6.3 d y d + dy d + y = y =, dy d = 3 a, b
Excel ではじめる数値解析 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 初版 1 刷発行時のものです.
Excel ではじめる数値解析 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/009631 このサンプルページの内容は, 初版 1 刷発行時のものです. Excel URL http://www.morikita.co.jp/books/mid/009631 i Microsoft Windows
x () g(x) = f(t) dt f(x), F (x) 3x () g(x) g (x) f(x), F (x) (3) h(x) = x 3x tf(t) dt.9 = {(x, y) ; x, y, x + y } f(x, y) = xy( x y). h (x) f(x), F (x
[ ] IC. f(x) = e x () f(x) f (x) () lim f(x) lim f(x) x + x (3) lim f(x) lim f(x) x + x (4) y = f(x) ( ) ( s46). < a < () a () lim a log xdx a log xdx ( ) n (3) lim log k log n n n k=.3 z = log(x + y ),
1 8, : 8.1 1, 2 z = ax + by + c ax by + z c = a b +1 x y z c = 0, (0, 0, c), n = ( a, b, 1). f = n i=1 a ii x 2 i + i<j 2a ij x i x j = ( x, A x), f =
1 8, : 8.1 1, z = ax + by + c ax by + z c = a b +1 x y z c = 0, (0, 0, c), n = ( a, b, 1). f = a ii x i + i
S I. dy fx x fx y fx + C 3 C vt dy fx 4 x, y dy yt gt + Ct + C dt v e kt xt v e kt + C k x v k + C C xt v k 3 r r + dr e kt S Sr πr dt d v } dt k e kt
S I. x yx y y, y,. F x, y, y, y,, y n http://ayapin.film.s.dendai.ac.jp/~matuda n /TeX/lecture.html PDF PS yx.................................... 3.3.................... 9.4................5..............
gnuplot.dvi
gnuplot gnuplot 1 gnuplot exit 2 10 10 2.1 2 plot x plot sin(x) plot [-20:20] sin(x) plot [-20:20][0.5:1] sin(x), x, cos(x) + - * / ** 5 ** plot 2**x y =2 x sin(x) cos(x) exp(x) e x abs(x) log(x) log10(x)
2.2 Sage I 11 factor Sage Sage exit quit 1 sage : exit 2 Exiting Sage ( CPU time 0m0.06s, Wall time 2m8.71 s). 2.2 Sage Python Sage 1. Sage.sage 2. sa
I 2017 11 1 SageMath SageMath( Sage ) Sage Python Sage Python Sage Maxima Maxima Sage Sage Sage Linux, Mac, Windows *1 2 Sage Sage 4 1. ( sage CUI) 2. Sage ( sage.sage ) 3. Sage ( notebook() ) 4. Sage
Solutions to Quiz 1 (April 20, 2007) 1. P, Q, R (P Q) R Q (P R) P Q R (P Q) R Q (P R) X T T T T T T T T T T F T F F F T T F T F T T T T T F F F T T F
Quiz 1 Due at 10:00 a.m. on April 20, 2007 Division: ID#: Name: 1. P, Q, R (P Q) R Q (P R) P Q R (P Q) R Q (P R) X T T T T T T F T T F T T T F F T F T T T F T F T F F T T F F F T 2. 1.1 (1) (7) p.44 (1)-(4)
I
I 6 4 10 1 1 1.1............... 1 1................ 1 1.3.................... 1.4............... 1.4.1.............. 1.4................. 1.4.3........... 3 1.4.4.. 3 1.5.......... 3 1.5.1..............
数学の基礎訓練I
I 9 6 13 1 1 1.1............... 1 1................ 1 1.3.................... 1.4............... 1.4.1.............. 1.4................. 3 1.4.3........... 3 1.4.4.. 3 1.5.......... 3 1.5.1..............
[ ] 0.1 lim x 0 e 3x 1 x IC ( 11) ( s114901) 0.2 (1) y = e 2x (x 2 + 1) (2) y = x/(x 2 + 1) 0.3 dx (1) 1 4x 2 (2) e x sin 2xdx (3) sin 2 xdx ( 11) ( s
![[ ] 0.1 lim x 0 e 3x 1 x IC ( 11) ( s114901) 0.2 (1) y = e 2x (x 2 + 1) (2) y = x/(x 2 + 1) 0.3 dx (1) 1 4x 2 (2) e x sin 2xdx (3) sin 2 xdx ( 11) ( s](/thumbs/94/118579915.jpg "[ ] 0.1 lim x 0 e 3x 1 x IC ( 11) ( s114901) 0.2 (1) y = e 2x (x 2 + 1) (2) y = x/(x 2 + 1) 0.3 dx (1) 1 4x 2 (2) e x sin 2xdx (3) sin 2 xdx ( 11) ( s")
[ ]. lim e 3 IC ) s49). y = e + ) ) y = / + ).3 d 4 ) e sin d 3) sin d ) s49) s493).4 z = y z z y s494).5 + y = 4 =.6 s495) dy = 3e ) d dy d = y s496).7 lim ) lim e s49).8 y = e sin ) y = sin e 3) y =
2 (1) a = ( 2, 2), b = (1, 2), c = (4, 4) c = l a + k b l, k (2) a = (3, 5) (1) (4, 4) = l( 2, 2) + k(1, 2), (4, 4) = ( 2l + k, 2l 2k) 2l + k = 4, 2l
ABCDEF a = AB, b = a b (1) AC (3) CD (2) AD (4) CE AF B C a A D b F E (1) AC = AB + BC = AB + AO = AB + ( AB + AF) = a + ( a + b) = 2 a + b (2) AD = 2 AO = 2( AB + AF) = 2( a + b) (3) CD = AF = b (4) CE
Contents 1 Scilab
Scilab (Shuji Yoshikawa) December 18, 2017 Contents 1 Scilab 3 1.1..................................... 3 1.2..................................... 3 1.3....................................... 3 1.4............................
all.dvi
fortran 1996 4 18 2007 6 11 2012 11 12 1 3 1.1..................................... 3 1.2.............................. 3 2 fortran I 5 2.1 write................................ 5 2.2.................................
7 27 7.1........................................ 27 7.2.......................................... 28 1 ( a 3 = 3 = 3 a a > 0(a a a a < 0(a a a -1 1 6
26 11 5 1 ( 2 2 2 3 5 3.1...................................... 5 3.2....................................... 5 3.3....................................... 6 3.4....................................... 7
5.. z = f(x, y) y y = b f x x g(x) f(x, b) g x ( ) A = lim h 0 g(a + h) g(a) h g(x) a A = g (a) = f x (a, b)
5 partial differentiation (total) differentiation 5. z = f(x, y) (a, b) A = lim h 0 f(a + h, b) f(a, b) h............................................................... ( ) f(x, y) (a, b) x A (a, b) x
.1 A cos 2π 3 sin 2π 3 sin 2π 3 cos 2π 3 T ra 2 deta T ra 2 deta T ra 2 deta a + d 2 ad bc a 2 + d 2 + ad + bc A 3 a b a 2 + bc ba + d c d ca + d bc +
.1 n.1 1 A T ra A A a b c d A 2 a b a b c d c d a 2 + bc ab + bd ac + cd bc + d 2 a 2 + bc ba + d ca + d bc + d 2 A a + d b c T ra A T ra A 2 A 2 A A 2 A 2 A n A A n cos 2π sin 2π n n A k sin 2π cos 2π
.3. (x, x = (, u = = 4 (, x x = 4 x, x 0 x = 0 x = 4 x.4. ( z + z = 8 z, z 0 (z, z = (0, 8, (,, (8, 0 3 (0, 8, (,, (8, 0 z = z 4 z (g f(x = g(
06 5.. ( y = x x y 5 y 5 = (x y = x + ( y = x + y = x y.. ( Y = C + I = 50 + 0.5Y + 50 r r = 00 0.5Y ( L = M Y r = 00 r = 0.5Y 50 (3 00 0.5Y = 0.5Y 50 Y = 50, r = 5 .3. (x, x = (, u = = 4 (, x x = 4 x,
Chap10.dvi
=0. f = 2 +3 { 2 +3 0 2 f = 1 =0 { sin 0 3 f = 1 =0 2 sin 1 0 4 f = 0 =0 { 1 0 5 f = 0 =0 f 3 2 lim = lim 0 0 0 =0 =0. f 0 = 0. 2 =0. 3 4 f 1 lim 0 0 = lim 0 sin 2 cos 1 = lim 0 2 sin = lim =0 0 2 =0.
IMO 1 n, 21n n (x + 2x 1) + (x 2x 1) = A, x, (a) A = 2, (b) A = 1, (c) A = 2?, 3 a, b, c cos x a cos 2 x + b cos x + c = 0 cos 2x a
1 40 (1959 1999 ) (IMO) 41 (2000 ) WEB 1 1959 1 IMO 1 n, 21n + 4 13n + 3 2 (x + 2x 1) + (x 2x 1) = A, x, (a) A = 2, (b) A = 1, (c) A = 2?, 3 a, b, c cos x a cos 2 x + b cos x + c = 0 cos 2x a = 4, b =
y π π O π x 9 s94.5 y dy dx. y = x + 3 y = x logx + 9 s9.6 z z x, z y. z = xy + y 3 z = sinx y 9 s x dx π x cos xdx 9 s93.8 a, fx = e x ax,. a =
[ ] 9 IC. dx = 3x 4y dt dy dt = x y u xt = expλt u yt λ u u t = u u u + u = xt yt 6 3. u = x, y, z = x + y + z u u 9 s9 grad u ux, y, z = c c : grad u = u x i + u y j + u k i, j, k z x, y, z grad u v =
B. 41 II: 2 ;; 4 B [ ] S 1 S 2 S 1 S O S 1 S P 2 3 P P : 2.13:
![B. 41 II: 2 ;; 4 B [ ] S 1 S 2 S 1 S O S 1 S P 2 3 P P : 2.13:](/thumbs/94/118646867.jpg "B. 41 II: 2 ;; 4 B [ ] S 1 S 2 S 1 S O S 1 S P 2 3 P P : 2.13:")
B. 41 II: ;; 4 B [] S 1 S S 1 S.1 O S 1 S 1.13 P 3 P 5 7 P.1:.13: 4 4.14 C d A B x l l d C B 1 l.14: AB A 1 B 0 AB 0 O OP = x P l AP BP AB AP BP 1 (.4)(.5) x l x sin = p l + x x l (.4)(.5) m d A x P O
f(x) = f(x ) + α(x)(x x ) α(x) x = x. x = f (y), x = f (y ) y = f f (y) = f f (y ) + α(f (y))(f (y) f (y )) f (y) = f (y ) + α(f (y)) (y y ) ( (2) ) f
22 A 3,4 No.3 () (2) (3) (4), (5) (6) (7) (8) () n x = (x,, x n ), = (,, n ), x = ( (x i i ) 2 ) /2 f(x) R n f(x) = f() + i α i (x ) i + o( x ) α,, α n g(x) = o( x )) lim x g(x) x = y = f() + i α i(x )
S I. dy fx x fx y fx + C 3 C dy fx 4 x, y dy v C xt y C v e kt k > xt yt gt [ v dt dt v e kt xt v e kt + C k x v + C C k xt v k 3 r r + dr e kt S dt d
S I.. http://ayapin.film.s.dendai.ac.jp/~matuda /TeX/lecture.html PDF PS.................................... 3.3.................... 9.4................5.............. 3 5. Laplace................. 5....
1 (1) ( i ) 60 (ii) 75 (iii) 315 (2) π ( i ) (ii) π (iii) 7 12 π ( (3) r, AOB = θ 0 < θ < π ) OAB A 2 OB P ( AB ) < ( AP ) (4) 0 < θ < π 2 sin θ
1 (1) ( i ) 60 (ii) 75 (iii) 15 () ( i ) (ii) 4 (iii) 7 1 ( () r, AOB = θ 0 < θ < ) OAB A OB P ( AB ) < ( AP ) (4) 0 < θ < sin θ < θ < tan θ 0 x, 0 y (1) sin x = sin y (x, y) () cos x cos y (x, y) 1 c
di-problem.dvi
005/05/05 by. I : : : : : : : : : : : : : : : : : : : : : : : : :. II : : : : : : : : : : : : : : : : : : : : : : : : : 3 3. III : : : : : : : : : : : : : : : : : : : : : : : : 4 4. : : : : : : : : : :
GNUPLOT 28 3 15 UNIX Microsoft-Windows GNUPLOT 4 GNUPLOT 1 GNUPLOT 2 2 3 2.1 UNIX.......................................... 3 2.2 Windows........................................ 4 2.3..................................
17 ( ) II III A B C(100 ) 1, 2, 6, 7 II A B (100 ) 2, 5, 6 II A B (80 ) 8 10 I II III A B C(80 ) 1 a 1 = 1 2 a n+1 = a n + 2n + 1 (n = 1,
17 ( ) 17 5 1 4 II III A B C(1 ) 1,, 6, 7 II A B (1 ), 5, 6 II A B (8 ) 8 1 I II III A B C(8 ) 1 a 1 1 a n+1 a n + n + 1 (n 1,,, ) {a n+1 n } (1) a 4 () a n OA OB AOB 6 OAB AB : 1 P OB Q OP AQ R (1) PQ
5.. z = f(x, y) y y = b f x x g(x) f(x, b) g x ( ) A = lim h g(a + h) g(a) h g(x) a A = g (a) = f x (a, b)............................................
5 partial differentiation (total) differentiation 5. z = f(x, y) (a, b) A = lim h f(a + h, b) f(a, b) h........................................................... ( ) f(x, y) (a, b) x A (a, b) x (a, b)
() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi)
0. A A = 4 IC () det A () A () x + y + z = x y z X Y Z = A x y z ( 5) ( s5590) 0. a + b + c b c () a a + b + c c a b a + b + c 0 a b c () a 0 c b b c 0 a c b a 0 0. A A = 7 5 4 5 0 ( 5) ( s5590) () A ()
1 1 3 ABCD ABD AC BD E E BD 1 : 2 (1) AB = AD =, AB AD = (2) AE = AB + (3) A F AD AE 2 = AF = AB + AD AF AE = t AC = t AE AC FC = t = (4) ABD ABCD 1 1
ABCD ABD AC BD E E BD : () AB = AD =, AB AD = () AE = AB + () A F AD AE = AF = AB + AD AF AE = t AC = t AE AC FC = t = (4) ABD ABCD AB + AD AB + 7 9 AD AB + AD AB + 9 7 4 9 AD () AB sin π = AB = ABD AD
1 12 ( )150 ( ( ) ) x M x 0 1 M 2 5x 2 + 4x + 3 x 2 1 M x M 2 1 M x (x + 1) 2 (1) x 2 + x + 1 M (2) 1 3 M (3) x 4 +
( )5 ( ( ) ) 4 6 7 9 M M 5 + 4 + M + M M + ( + ) () + + M () M () 4 + + M a b y = a + b a > () a b () y V a () V a b V n f() = n k= k k () < f() = log( ) t dt log () n+ (i) dt t (n + ) (ii) < t dt n+ n
num2.dvi
kanenko@mbk.nifty.com http://kanenko.a.la9.jp/ 16 32...... h 0 h = ε () 0 ( ) 0 1 IEEE754 (ieee754.c Kerosoft Ltd.!) 1 2 : OS! : WindowsXP ( ) : X Window xcalc.. (,.) C double 10,??? 3 :, ( ) : BASIC,
1/1 lim f(x, y) (x,y) (a,b) ( ) ( ) lim limf(x, y) lim lim f(x, y) x a y b y b x a ( ) ( ) xy x lim lim lim lim x y x y x + y y x x + y x x lim x x 1
1/5 ( ) Taylor ( 7.1) (x, y) f(x, y) f(x, y) x + y, xy, e x y,... 1 R {(x, y) x, y R} f(x, y) x y,xy e y log x,... R {(x, y, z) (x, y),z f(x, y)} R 3 z 1 (x + y ) z ax + by + c x 1 z ax + by + c y x +
高等学校学習指導要領解説 数学編
5 10 15 20 25 30 35 5 1 1 10 1 1 2 4 16 15 18 18 18 19 19 20 19 19 20 1 20 2 22 25 3 23 4 24 5 26 28 28 30 28 28 1 28 2 30 3 31 35 4 33 5 34 36 36 36 40 36 1 36 2 39 3 41 4 42 45 45 45 46 5 1 46 2 48 3
04.dvi
22 I 4-4 ( ) 4, [,b] 4 [,b] R, x =, x n = b, x i < x i+ n + = {x,,x n } [,b], = mx{ x i+ x i } 2 [,b] = {x,,x n }, ξ = {ξ,,ξ n }, x i ξ i x i, [,b] f: S,ξ (f) S,ξ (f) = n i= f(ξ i )(x i x i ) 3 [,b] f:,
さくらの個別指導 ( さくら教育研究所 ) A 2 P Q 3 R S T R S T P Q ( ) ( ) m n m n m n n n
1 1.1 1.1.1 A 2 P Q 3 R S T R S T P 80 50 60 Q 90 40 70 80 50 60 90 40 70 8 5 6 1 1 2 9 4 7 2 1 2 3 1 2 m n m n m n n n n 1.1 8 5 6 9 4 7 2 6 0 8 2 3 2 2 2 1 2 1 1.1 2 4 7 1 1 3 7 5 2 3 5 0 3 4 1 6 9 1
IA 2013 : :10722 : 2 : :2 :761 :1 (23-27) : : ( / ) (1 /, ) / e.g. (Taylar ) e x = 1 + x + x xn n! +... sin x = x x3 6 + x5 x2n+1 + (
IA 2013 : :10722 : 2 : :2 :761 :1 23-27) : : 1 1.1 / ) 1 /, ) / e.g. Taylar ) e x = 1 + x + x2 2 +... + xn n! +... sin x = x x3 6 + x5 x2n+1 + 1)n 5! 2n + 1)! 2 2.1 = 1 e.g. 0 = 0.00..., π = 3.14..., 1
2009 I 2 II III 14, 15, α β α β l 0 l l l l γ (1) γ = αβ (2) α β n n cos 2k n n π sin 2k n π k=1 k=1 3. a 0, a 1,..., a n α a
009 I II III 4, 5, 6 4 30. 0 α β α β l 0 l l l l γ ) γ αβ ) α β. n n cos k n n π sin k n π k k 3. a 0, a,..., a n α a 0 + a x + a x + + a n x n 0 ᾱ 4. [a, b] f y fx) y x 5. ) Arcsin 4) Arccos ) ) Arcsin
1 29 ( ) I II III A B (120 ) 2 5 I II III A B (120 ) 1, 6 8 I II A B (120 ) 1, 6, 7 I II A B (100 ) 1 OAB A B OA = 2 OA OB = 3 OB A B 2 :
9 ( ) 9 5 I II III A B (0 ) 5 I II III A B (0 ), 6 8 I II A B (0 ), 6, 7 I II A B (00 ) OAB A B OA = OA OB = OB A B : P OP AB Q OA = a OB = b () OP a b () OP OQ () a = 5 b = OP AB OAB PAB a f(x) = (log
x, y x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 xy (x y) (x + y) xy (x y) (x y) ( x 2 + xy + y 2) = 15 (x y)
x, y x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 1 1977 x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 xy (x y) (x + y) xy (x y) (x y) ( x 2 + xy + y 2) = 15 (x y) ( x 2 y + xy 2 x 2 2xy y 2) = 15 (x y) (x + y) (xy
combine: combine(exp1 ) collect: collect(exp1,x) convert: convert(exp1,opt ) > convert(sin(x),exp); > convert(sinh(x),exp); > convert(exp(i*x),trig);
1 Maple exp Maple 1.1 1.1.1 solve( ), diff( ), int( ), 1: simplify: lhs, rhs: subs: expand: numer, denom: assume: factor: coeff: assuming: normal: nops, op assign: combine: about: collect: anames( user
0 1-4. 1-5. (1) + b = b +, (2) b = b, (3) + 0 =, (4) 1 =, (5) ( + b) + c = + (b + c), (6) ( b) c = (b c), (7) (b + c) = b + c, (8) ( + b)c = c + bc (9
1-1. 1, 2, 3, 4, 5, 6, 7,, 100,, 1000, n, m m m n n 0 n, m m n 1-2. 0 m n m n 0 2 = 1.41421356 π = 3.141516 1-3. 1 0 1-4. 1-5. (1) + b = b +, (2) b = b, (3) + 0 =, (4) 1 =, (5) ( + b) + c = + (b + c),
1 I
1 I 3 1 1.1 R x, y R x + y R x y R x, y, z, a, b R (1.1) (x + y) + z = x + (y + z) (1.2) x + y = y + x (1.3) 0 R : 0 + x = x x R (1.4) x R, 1 ( x) R : x + ( x) = 0 (1.5) (x y) z = x (y z) (1.6) x y =
18 5 10 1 1 1.1 1.1.1 P Q P Q, P, Q P Q P Q P Q, P, Q 2 1 1.1.2 P.Q T F Z R 0 1 x, y x + y x y x y = y x x (y z) = (x y) z x + y = y + x x + (y + z) = (x + y) + z P.Q V = {T, F } V P.Q P.Q T F T F 1.1.3